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Log-Sobolev-type inequalities for solutions to stationary Fokker–Planck–Kolmogorov equations

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Abstract

We prove that every probability measure \(\mu \) satisfying the stationary Fokker–Planck–Kolmogorov equation obtained by a \(\mu \)-integrable perturbation v of the drift term \(-x\) of the Ornstein–Uhlenbeck operator is absolutely continuous with respect to the corresponding Gaussian measure \(\gamma \) and for the density \(f=d\mu /d\gamma \) the integral of \(f |\log (f+1)|^\alpha \) against \(\gamma \) is estimated via \(\Vert v\Vert _{L^1(\mu )}\) for all \(\alpha <1/4\), which is a weakened \(L^1\)-analog of the logarithmic Sobolev inequality. This yields that stationary measures of infinite-dimensional diffusions whose drifts are integrable perturbations of \(-x\) are absolutely continuous with respect to Gaussian measures. A generalization is obtained for equations on Riemannian manifolds.

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Authors and Affiliations

  1. Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia, 119991

    V. I. Bogachev, A. V. Shaposhnikov & S. V. Shaposhnikov

  2. National Research University Higher School of Economics, ul. Usacheva 6, Moscow, Russia, 119048

    V. I. Bogachev & S. V. Shaposhnikov

Authors
  1. V. I. Bogachev
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  2. A. V. Shaposhnikov
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  3. S. V. Shaposhnikov
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Correspondence to V. I. Bogachev.

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Communicated by L. Ambrosio.

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Bogachev, V.I., Shaposhnikov, A.V. & Shaposhnikov, S.V. Log-Sobolev-type inequalities for solutions to stationary Fokker–Planck–Kolmogorov equations. Calc. Var. 58, 176 (2019). https://doi.org/10.1007/s00526-019-1625-x

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